Asymptotic arbitrage and large deviations

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1 Math Finan Econ 8) :3 49 DOI.7/s Asymptotic arbitrage and large deviations H. Föllmer W. Schachermayer Received: 3 May 7 / Accepted: April 8 / Published online: May 8 Springer-Verlag 8 Abstract ypical models of mathematical finance admit equivalent martingale measures up to any finite time horizon but not globally, and this means that arbitrage opportunities arise in the long run. In this paper, we derive explicit estimates for asymptotic arbitrage, and we show how they are related to large deviation estimates for the market price of risk. As a case study we consider a geometric Ornstein Uhlenbeck process. In this setting we also compute the optimal trading strategies and the resulting optimal growth rates of expected utility for all HARA utilities. Keywords Asymptotic arbitrage Utility maximization Large deviations Cost averaging JEL Classification G G C6 Introduction In this paper, we investigate the issue of optimizing terminal wealth by investing in a financial market, when the time horizon tends to infinity. We focus on the following issues: i) A better understanding of the features of the financial market S t ) t which insure exponential growth of the terminal wealth X of an investor in this market, for. W. Schachermayer gratefully acknowledges Financial support from the Austrian Science Fund FWF) under the grant P9456, from Vienna Science and echnology Fund WWF) under Grant MA3 and by the Christian Doppler Research Association CDG). H. Föllmer Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 99 Berlin, Germany W. Schachermayer B) Vienna University of echnology, Wiedner Hauptstrasse 8-/5, 4 Wien, Austria wschach@fam.tuwien.ac.at 3

2 4 H. Föllmer, W. Schachermayer ii) iii) Relating this question to the notion of asymptotic arbitrage, to the theory of large deviations, and to utility maximization with respect to the power utilities Ux) = xα α, for α ], [. Analyzing carefully the case study of a geometric Ornstein Uhlenbeck process and comparing it to the well-known situation of the Black Scholes model by explicitly calculating the optimal strategies for HARA-utility optimizers. We consider an R d -valued semi-martingale S = S t ) t modeling the price process of d risky assets with infinite horizon. he bond is assumed to be normalized by B t, i.e., we consider S in discounted terms. For a fixed finite horizon,let K ={H S) H H}, denote the set of attainable contingent claims, where H denotes the class of predictable, S-integrable, admissible processes for a definition see e.g. [7, Definition 8.., p. 3]). We recall the notion of strong asymptotic arbitrage as introduced by Kabanov and Kramkov [9] which we specialize to the present situation of a varying time horizon. Definition. he process S = S t ) t allows for strong asymptotic arbitrage if, for ε>, there is < and X K satisfying i) X ε, a.s., and ii) P[X ε ] ε. he economic interpretation is straightforward: condition i) means that the maximal loss of the trading strategy, yielding the wealth X at time, is bounded by ε; condition ii) means that with probability ε the terminal wealth X equals at least ε. In [9] a dual characterization of this concept was given in terms of the Hellinger distance of the equivalent martingale measures for the process S t ) t to the original measure P compare also []). In Proposition.we take up this theme again and relate these Hellinger distances to utility maximization for power utility Ux) = xα α,whereα ], [. For the sake of clarity of exposition we shall focus on the setting of diffusions driven by Brownian motion; we note however, that many of the results below could be extended to more general situations. Let, F,F t ) t, P) be a filtered probability space such that F t ) t is the right continuous, saturated) filtration generated by an R N -valued standard Brownian motion W t ) t. Assumption. he process S = S t ) t will be assumed to be an R d -valued diffusion based on, F,F t ) t, P) such that there is a deterministic, time-independent) volatility function as well as a market price of risk function such that σ : R d R d N ) ϕ : R d R N ) ds t = σs t )dw t + ϕs t )dt), 3) where we may and do suppose that ϕs t ) takes its values in kerσ S t )). 3

3 Asymptotic arbitrage and large deviations 5 We also assume that Z min t = exp t ϕ u, dw u ) t ϕ u du 4) is a strictly positive martingale, where we put ϕ u = ϕs u ) and where.,.)and. denote the inner product and the corresponding Euclidean norm on R N. We then deduce from Girsanov s formula that, for each, the measure Q min on F defined by dq min dp = Z min is a probability measure equivalent to the restriction of P to F such that S t ) t is a local martingale under Q min. Following [5] we call Qmin the minimal martingale measure for S t ) t. It seems worthwile to comment on the above concepts. Suppose for simplicity that ϕ and σ are constant. If the d N matrix σ is injective, then the law of the process S in 3) uniquely determines the vector ϕ R N ; this corresponds to the case of a complete market. On the other hand, in the incomplete case, i.e. for a non-injective matrix σ, the vector ϕ is only determined by the process S in 3) up to adding elements in the kernel of σ. here is one canonical choice of ϕ, namely the one orthogonal to kerσ ). he fact that this choice of ϕ has minimal norm in R N motivates the name minimal above. A central question of our present investigation will be to understand which features of the above model imply that there is exponential growth of a well chosen attainable portfolio as the time horizon goes to. It is obvious that one has to impose some assumption on the market price of risk. Indeed, if ϕ vanishes then the process S t ) t is a local martingale, and one cannot systematically win by betting on a local martingale in an admissible way. Definition.3 Under the above assumptions we say that the diffusion process S = S t ) t has an average squared market price of risk above the threshold c > if the process ϕ t ) t satisfies the following estimate: P ϕ t dt < c =. 5) If there is c > such that 5) holds true we say that S has a non-trivial market price of risk. We say that the market price of risk satisfies a large deviations estimate if there are constants c, c > such that sup log P ϕ t dt c < c. 6) he economic interpretation of 5) is that the market price of risk should on average be bounded away from zero in the long run. his assumption is satisfied whenever the diffusion is ergodic with invariant measure µ and if the market price of risk function ϕ is not µ-a.s. equal to, since ergodicity implies ϕ t dt = R d ϕx) µdx) P-a.s. 3

4 6 H. Föllmer, W. Schachermayer As explained in Sect. 3, the large deviations estimate 6) for the market price of risk follows by a contraction principle whenever the diffusion process S t ) t is ergodic and satisfies a principle of large deviations. We refer to the large deviations literature for suitable conditions; see, e.g., [8]. For the Black Scholes model the market price of risk is constant, and so the large deviations estimate 6) is trivially satisfied if the market price of risk is assumed to be different from zero. Less trivial is the example of the geometric Ornstein Uhlenbeck process which will be analyzed in Sects. 4 and 5, using large deviations results from [4,8]); see also [3]. In this example the large deviations estimate 6) holds true, and the optimal pairs c, c ) can be calculated explicitly. Assumption 5) is of course weaker than 6), but it is sufficient to deduce the following estimates which will be proved in Sect. 3. heorem.4 Let S = S t ) t be a process satisfying Assumption. and having an average squared market price of risk above the threshold c > ; cf.5). For ε>, γ + γ < c/, and for large enough, there exists X K such that i) X e γ, ii) P [ X e γ ] ε. he message of the theorem is that the assumption of a non-trivial market price of risk 5) implies asymptotic arbitrage; in fact we obtain exponential estimates for the maximal loss in i) as well as for the typical growth in ii). If Assumption 5) is replaced by the stronger large deviation estimate 6), one should even expect an exponential decay in time for the probability of falling short of the exponential lower bound in assertion ii) above. his will be discussed in Sect. 3. In Sect. 4 we illustrate the situation for the geometric Ornstein Uhlenbeck process defined by S t = expy t ), 7) where Y t ) t denotes an Ornstein Uhlenbeck process defined by dy t = ρy t dt + σ dw t, Y = y, 8) for constants σ>, ρ>, and y R. Here the market price of risk ϕ t = ϕs t ) is given by ϕ t = ρ σ Y t + σ see 4)), and we obtain the subsequent explicit results: heorem.5 Let S t ) t be a geometric ] Ornstein Uhlenbeck [ process as in 7). For any γ,γ,γ R such that γ + γ <γ, σ 8 + ρ 4, there exist attainable contingent claims X K such that i) X e γ, ii) log P [ X < e γ ] ) = σ 8 γ + ρ 4. σ 8 γ + ρ In fact we are going to prove a stronger version, with sharper bounds in i) and a corresponding optimality result for the rate of convergence in ii); cf. heorem 4. and Remark 4.3. Remark 4.4 describes the extension to an Ornstein Uhlenbeck process with drift. A corresponding result for the Black Scholes model is given in heorem

5 Asymptotic arbitrage and large deviations 7 How are these results, which ensure asymptotic arbitrage in a rather strong sense, related to the theme of utility maximization? For the case of logarithmic utility, heorem.4 easily implies a lower bound for the attainable expected utility per unit time; see Proposition 3.. In fact, there is a deeper connection of the results on asymptotic arbitrage and utility maximization. In Proposition.3 we give rather sharp estimates between asymptotic arbitrage and utility maximization with respect to the power utilities Ux) = xα α,whereα ranges in ], [. his result is interesting in its own right and also allows for a better understanding of the duality theory of asymptotic arbitrage [9,]. In Sect. 5 we continue our case study of the geometric Ornstein Uhlenbeck process and calculate explicitly the optimal trading strategies and the resulting expected utility u x) for all > and all HARA-utilities. As a corollary we obtain the optimal growth rates for u x). In the case of power utility, this may be viewed as a probabilistic complement to the dynamic programming approach in Fleming and Sheu [3], as explained in Remark 5.8. While for logarithmic utility there are no surprises and our findings are similar to the wellknown results of R. Merton in the case of the Black Scholes model [6,7], we find some counter-intuitive results in the case of power utility Ux) = xα α,forα ], [ \{}, and of exponential utility Ux) = exp λx). Asymptotic arbitrage and power utility We start with an easy but surprisingly sharp dual characterization of the notion of asymptotic arbitrage. In this proposition we take the horizon as fixed. Proposition. Let S = S t ) t be an R d -valued locally bounded semi-martingale such that M e S) ={Q P SisalocalQ-martingale} =. For >ε,ε > the statements a) and b) are equivalent: a) here is X K such that i) X ε, ii) P[X ε ] ε. b) here is A F with P[A ] ε such that, for each Q M e S) we have Q[A ] ε. In this case we say that S admits an ε,ε )-arbitrage up to time ). Proof a) b): Let A ={X < ε } so that P[A ] ε,andtakeq M e S). Since X K implies E Q [X ] see, e.g., [5, heorem 5.7]), we may estimate ε )Q[ \A ] ε Q[A ] E Q [X ], hence Q[A ] ε. b) a): If A satisfies the assumptions of b) then X = ε A + ε ) \A has properties i) and ii) and satisfies E Q [X ] for each Q M e S). Applying again the superhedging theorem [5, heorem 5.7] we see that X is dominated by an element X K,and X clearly inherits properties i) and ii). 3

6 8 H. Föllmer, W. Schachermayer We remark that the assumption of local boundedness is not really relevant in the present context. It could be dropped by replacing the concept of local martingales by the concept of sigma-martingales see [6]). In the next result we relate strong asymptotic arbitrage with dynamic portfolio optimisation for a certain class of utility functions which includes the power utilities Ux) = xα α for <α<astypical examples. Proposition. Let S = S t ) t be an R d -valued locally bounded semi-martingale such that, for each >, thesetm e S) is not empty. Let U : R + R be a strictly increasing concave function such that Ux) =, x Ux) =. x he following assertions are equivalent: a) b) S t ) t allows for strong asymptotic arbitrage. Defining the value function we have u x) = sup E[Ux + X )], X K u x) =, for some x > or, equivalently, for all x > ). Proof a) b): Given x >, let < ε < x.ifx Definition. we get K satisfies i) and ii) of u x) E[Ux + X )] x ) U P[X <ε ]+Uε )P[X ε ] x ) εu + Uε ). Since the latter expression tends to zero as ε, we obtain u x) =. b) a): Suppose that there is x > such that u x) =. hus we may find X K such that X x and E[Ux + X )] =. ake ε>. Applying schebyscheff s inequality in the form we obtain ε > such that P[X < xε ]=P[ Ux + X ) > Ux + xε ) ] Ux + xε )] E[Ux + X )], P[X xε ] ε for ε.sincex x we conclude that εx /x satisfies the requirements of Definition.. 3

7 Asymptotic arbitrage and large deviations 9 he preceding result is of a purely qualitative nature. Since we also want to obtain quantitative results on the speed of convergence, we now specialize to the case of power utility U α) x) = xα α, In this case the conjugate function <α<. is given by V y) = sup[u α) x) xy], y >, x> α V y) = V β) y) = yβ β where β = α ], [. As usual in utility optimization we write for the primal and dual value functions [4] u α) x) = sup E[U α) x + X )], x >, X K [ )] v β) y) = inf E V β) y dq Q M e S) dp, y >. We obtain from the scaling property of the power function that we have u α) x) = c U α) x), for some c, as well as v β) y) = c V β) y), wherec = c α + compare [4]). he following proposition is similar to the results in [9] in terms of the concept of Hellinger distance, but there it is not connected with the idea of utility maximization. Again the horizon will be fixed. Proposition.3 Let <α<, β = α α, and an Rd -valued semi-martingale S = S t ) t as in Proposition. be given. For ε, ε,ε > consider the following statements. i) S admits an ε,ε )-arbitrage. i ) here is A F, P[A] ε, such that, for each Q M e S) we have Q[A] ε. ii) For each Q M e S) there is A F, such that P[A] ε and Q[A] ε. iii) u α) x) εu α) x), for some or, equivalently, for all) x >. iii ) v β) y) ε α + V β) y), for some or, equivalently, for all) y >. hen the following assertions hold true: a) b) c) d) i) i ) and iii) iii ). i ) ii). he reverse implication ii) i ) holds true if M e S) ={Q} is a singleton complete financial market). ii) iii) if ε + α maxε,ε α ). iii) i) if ε ε ε α. Proof a) follows from Proposition. and the discussion preceding Proposition.3, while b) is obvious. c): Fix Q M e S) and the corresponding set A F satisfying P[A] ε and Q[A] ε as in ii). By possibly passing to smaller values of ε,ε, we may suppose 3

8 H. Föllmer, W. Schachermayer that P[A] =ε and Q[A] = ε. In view of a) it is enough to verify iii ). By Jensen we obtain [ )] E V β) dq dp = [ ) ] β β E dq dp [ ) Q[A] β ) ] Q[ \ A] β β E A + \A P[A] [ ε β ε ) β + ε ) β ε β = β ) β max ε β,ε β. As this inequality holds true for all Q M e S) we obtain ) v β) ) max ε β,ε β V β) ). Using that α + = α = β we conclude from the assumption ) ε α + max α + α α + ε,ε ) = max ε β,ε β P[ \ A] ] v β) ) ε α + V β) ), which yields iii ). d): Let ε, ε,ε satisfy ε<ε ε α. By assumption iii) there is X K such that E[U α) ε + X )] εu α) ε ) = ε εα α. Clearly X ε a.s. In order to verify i) it remains to show that P[X ε ] ε. Indeed, using schebyscheff we obtain E[U α) ε + X )] P[X < ε ] U α) ), hence and therefore ε εα α P[X < ε ] ), α ε > P[X < ε ]. Remark.4 Statements i ) and ii) above only differ in the order of the quantifiers, so that we have the trivial implication i ) ii), as well as ii) i ) in the case when M e S) is a singleton. he more interesting part of Proposition.3 is that we may also conclude that ii) i ) in the incomplete case, provided we replace the constants ε,ε in i ) by bigger constants 3

9 Asymptotic arbitrage and large deviations ε, ε. For example we may choose ε = + α ε, ε = ε,whereα ], [ satisfies ε = ε α. Indeed, for given ε,ε > andα ], [ such that ε = ε α,wehave ii) iii ), if we let ε = + α ε.wethenhaveε ε ε α = + α ε ε so that we obtain iii) i) i ), if we use in i) and i ) the constants ε, ε instead of ε,ε. Summing up, we can reverse the quantifiers in statement ii) provided we content ourselves with somewhat worse constants. his phenomenon is related to a quantitative version of the Halmos Savage theorem, as was stressed in []. 3 Estimates for asymptotic arbitrage In this section we show under suitable regularity conditions that price processes with a nontrivial market price of risk 5) allow for asymptotic arbitrage; more precisely, we prove the estimates of heorem.4. We also show how these estimates can be refined if the market price of risk satisfies a large deviations estimate. Let us fix a diffusion process S satisfying Assumption., and recall the minimal martingale measure Q min for S t ) t defined via 4). It follows from [5] that, for an arbitrary Q M e S) there is a predictable process ψ t ) t such that the density Z = dq dp is given by Z = exp ψ dt) t dw t ψ t, 9) where ψ t ϕ t is in the kernel of σs t ) a.s. for almost all t ). Hence ϕ t is orthogonal to ψ t ϕ t,sothat ψ t ϕ t. herefore the estimates 5)and6) carry over from ϕ t to ψ t. Proposition 3. Suppose that S t ) t satisfies Assumption. and has an average squared market price of risk above the threshold c > ;cf.5). For ε>and <γ < c there is > such that, for, condition ii) of Proposition.3 is satisfied with ε = ε and ε = e γ, i.e., for each Q M e S) there is A F such that P[A ] ε and Q[A ] e γ. Proof Fix <γ <γ< c and find > 4γ γ γ) ε such that, for, P ϕ t dt γ < ε. ) For Q M e S) let ψ t) t be the R N -valued process as in 9) sothat dq dp = Z = exp ψ dt) t dw t ψ t. Define the stopping time τ by τ = inf t [, ] t ψ s ds γ. 3

10 H. Föllmer, W. Schachermayer For the random variable B τ = τ ψ t dw t we infer from τ ψ t dt γ that B τ L γ, P) and so schebyscheff s inequality implies For P [ B τ γ γ) ] Z τ = exp τ γ γ γ) < ε. ) ψ dt) t dw t ψ t we obtain from )and)that P [ Z τ > exp γ ) ] τ ψ t = P B τ dt > γ P [ B τ γ γ) ] τ + P ψ t dt < γ ε + ε. Letting A = {Z τ > exp γ )} we obtain P[A ] ε and Q [A c ]=E[Z τ A c ] e γ. We are now in a position to construct attainable contingent claims which satisfy the arbitrage estimates of heorem.4. Proof of heorem.4 Let the constant c > begivenby5). he preceding Proposition 3. implies that, for any given ε > andε = e d,where < d < c, condition ii) of Proposition.3 is satisfied for sufficiently large. Fix ε>andγ,γ,γ verifying <γ = γ + γ < c, as in the statement of heorem.4.fixd > with γ<d < c,and let µ = d γ d ], [. We now proceed similarly as in Remark.4. Choose such that for > condition ii) of Proposition.3 is satisfied with ε ) µ ε = and ε = e d. 3 Defining α ], [ by the equation ε = ε α. we have that α as. We may assume that has been chosen large enough such that + α 3, for. 3

11 Asymptotic arbitrage and large deviations 3 Letting ε = 3ε µ = ε and ε = ε µ) we have ε ε α = 3ε µ ε µ) α = 3ε + α max ε,ε α he last inequality implies that condition iii) of Proposition.3 is satisfied with ε replaced by 3ε. Hence, we may conclude that condition i) of Proposition.3 is satisfied, i.e., there is an ε, ε )-arbitrage, for the pair ε, ε ) = ε, e d µ) ). Note that d µ) > γ. Hence, there is X K such that i) X e d µ) a.s. ii) P [ X e d µ) ] ε. Since d µ) γ >γ, we see that the contingent claim X = e d µ) γ ) X K satisfies, for sufficiently large, ). i) X e γ a.s., ii) P[X e γ ] ε. he message of heorem.4 is that we may achieve exponential growth of a portfolio X with probability close to as, while also controlling the maximal loss by an exponential bound. his implies, for instance, the following lower bound for the growth of logarithmic utility. Corollary 3. Let S = S t ) t be a process satisfying Assumption. and having an average squared market price of risk above the threshold c > as in 5). For any initial capital x >, there exist contingent claims X K such that inf E [ logx + X ) ] c. Proof By heorem.4 we know that for γ = andγ < c/ we may find, for ε>and sufficiently large, X K with i) X, ii) P [ X e γ ] ε. For <α<x we obtain the estimate E[logx + αx )] ε logx α) + ε) logx + αeγ ), hence E[logx + αx )] ε)γ, for sufficiently large, which readily yields the result. heorem.4 is valid under the Assumption 5) of a non-trivial market price of risk. If we replace this assumption by the stronger large deviation estimate 6) we expect a stronger result: the term P[X < e γ ] in statement ii) of heorem.4 should even decay exponentially as. Let us sketch a systematic approach how to obtain such a sharper result; we also take the opportunity to motivate the large deviation estimate 6). 3

12 4 H. Föllmer, W. Schachermayer Let S t ) t be a d-dimensional diffusion satisfying Assumption., and assume that S is in fact ergodic. hus, the empirical distributions ρ ω) := δ St ω)dt converge weakly, P-a.s., to the unique invariant distribution µ of S. ypically, the empirical distributions will satisfy a large deviations principle of the form log P[ρ A] inf I ν) ) ν A with some rate function I defined on the convex set M R d ) of probability measures on R d, where ) should be read as an upper bound for the sup of the left-hand side if A is a closed subset of M R d ) and as a lower bound for the inf if A is open. Under some regularity conditions the market price of risk will then satisfy the large deviations estimate 6) log P ϕ t dt c c for any c < ϕx) µdx), with c := inf I ν) ϕx) dνx) c >, due to the contraction principle see, e.g., [8]). In such a situation one should expect an exponential decay of the probabilities P [Z ] e γ and a corresponding exponential version of the estimates in heorem.4, assuming now uniqueness of the locally equivalent martingale measure Q. However, this will involve a slight refinement of the large deviations principle in its classical form ). Let us sketch the argument for the one-dimensional case. Indeed, By Itô s formula, 3 [ ] P Z e γ = P ϕ t dw t := = ) dst ϕ t σs t ) ϕ tdt f S t )ds t = FS ) FS ) ϕ t dw t + ϕ t dt ϕt dt γ. f S t )σ S t )dt ϕ S t )dt,

13 Asymptotic arbitrage and large deviations 5 whereweput f x) := ϕx) σx), assume f C,andtakeF C such that F = f. hus, [ ] P Z e γ = P FS ) FS )) hs t )dt γ = P FS ) FS )) hx)dρ x) γ wherewedefinehx) = f x)σ x) + ϕ x). An exponential estimate of the form [ ] sup log P Z e γ < will now follow for γ< hdµ, if we have a joint large deviation principle for the random variables FS ) FS ) ) ),ρ with values in R M R d ) or, more specifically, for the random variables FS ) FS )) hx)dρ x). In this paper, we do not try to give a rigorous version of the argument under general conditions. Instead we will illustrate this approach by proving explicit exponential estimates for the geometric Ornstein Uhlenbeck process. 4 A case study: the geometric Ornstein Uhlenbeck process In this section we consider the geometric Ornstein Uhlenbeck process and derive exponential versions of the extimates in heorem.4. In order to keep the notation as transparent as possible, we take the simplest case S t = expy t ), 3) where Y t ) t is the stationary Ornstein Uhlenbeck process defined by dy t = ρy t dt + σ dw t, Y = y, with parameters ρ>andσ>and with initial value y R; see, however, Remark 4.4 below for the extension to an Ornstein Uhlenbeck process with drift. he process S = S t ) t defined by 3) satisfies the SDE σ ds t = S t [ ρy t dt + σ dw t ]+S t dt [ = S t σ dw t ρy t σ ) ] dt. 4) σ For fixed >, the unique equivalent martingale measure Q on F for the process S is defined by dq dp F = Z 3

14 6 H. Föllmer, W. Schachermayer where Z := exp σ ρy t σ ) dw t ρy σ t σ ) dt. 5) In other words, the dynamics of S under Q takes the form where ds t = σ S t dw Q t, W Q t t := W t = σ σ ρy s σ Y t y + σ ) t ) ds 6) is a Wiener process under Q. In this specific model we can describe the large deviations more explicitly. Proposition 4. For any γ and ] [, σ 8 + ρ 4,thesetsA := { Z e γ } satisfy Q[A ] e γ 7) σ log P[A 8 γ + ρ 4 ]= σ 8 γ + ρ ). 8) Proof Clearly, [ ] Q Z e γ = Z dp e γ, {Z e γ } and this implies 7). he proof of 8) consists in combining a result of Florens Landais and Pham [4] for certain large deviations of the Ornstein Uhlenbeck process with a simple perturbation argument. o this end, we write log Z = σ 3 = ρ σ ρy t σ = ρ σ ξ σ ) σ dy t + ρ ) σ Y tdt ρ σ Y t dy t Y Y ) + ρ σ Yt dt σ 8 8, Yt dt σ 8 + ρ Y t dt 9)

15 Asymptotic arbitrage and large deviations 7 where ξ := η + ζ, η := Y t dy t + ρ Y t dt = Y Y σ ) + ρ ζ := σ ρ Y Y ). Y t dt, Recall that the Ornstein Uhlenbeck process Y is ergodic with invariant distribution N, σ ρ ), and note that this implies ξ = η = σ + ρ σ ρ = σ 4 P-a.s. ) Let us now look at the corresponding large deviations. Applying heorem. of Florens Landais and Pham [4] with θ = ρ, θ = ρ, and the straightforward extension to the case σ = ), we see that the random variables η satisfy a large deviations principle with rate function ) ρ y σ I y) = + 4 y σ + for y > σ for y σ i.e., sup log P [ η F ] inf y F I y) for any closed set F R and inf log P [ η G ] inf y G I y) for any open set G R. A simple perturbation argument shows that the same large deviations principle with rate function I applies to the random variables ξ. Indeed, the additional terms ζ are normally distributed with means m converging to m R and variances σ converging to σ ], [, and this implies [ ] log P ζ >ε = = ε σ. log π ε σ e y dy 3

16 8 H. Föllmer, W. Schachermayer In particular, [ ] [ ] ξ P F ξ [ P >ε η ] + P F ε [ η ] P F ε for any ε>, F ε := {y dy, F) ε}, andfor large enough. hus, [ ] log P ξ F inf I y) y F ε for closed F and for arbitrary ε>, hence [ ] log P ξ F inf y F I y). he lower bound for open sets follows in the same manner. For y > σ 4 and F := [y, [, these two bounds imply [ ] log P ξ y = I y ). We can now conclude that the asymptotic behavior of [ ] [ ] ρ P Z e γ = P for γ σ ξ 8 σ γ σ )] 8 γ [ ξ = P σ ρ ] [, σ 8 + ρ 4 is described by Eq. 8). We now are prepared to prove heorem.5. In fact we will prove the following stronger version: heorem ] 4. Let [ S t ) t be the geometric Ornstein Uhlenbeck process defined in 7).ake γ, σ 8 + ρ 4 and any γ <γ. hen there exist contingent claims X K such that, for any γ <γ γ, i) X e γ for large, ii) log P [ X < e γ ] = ) σ 8 γ + ρ 4 σ 8 γ + ρ More precisely, we can achieve the tighter bounds iii) X α := e γ Q[Ac ] Q[A ] where A := {Z e γ }, and the decay rate in ii) for the shortfall probabilities P[X < e γ ] is in fact optimal under the constraint iii). Proof Since Q[A c ] e γ by 7), the constants α defined by iii) satisfy α e γ γ ) Q[A ], 3.

17 Asymptotic arbitrage and large deviations 9 hence α e γ for large if γ <γ γ. Now consider the contingent claims Clearly, X α,and hence X K. Moreover, since X := e γ A c α A. E Q [X ]=e γ Q[A c ] α Q[A ]=, { X < e γ } = A for large enough, part ii) of heorem 4. follows from 8). In order to check the optimality of the convergence rate in ii) under the constraint iii), take a sequence of contingent claims X K such that X α. he corresponding sets à := { X < e γ } satify Q [ à c ] ) e γ + α α = e γ Q [ à c ] + α )Q [ ] à ] E Q [ X, hence Q [ à c ] α α + e γ = Q[Ac ]. he Neyman Pearson lemma allows us to conclude that P [ à c ] P[A c ]. In particular, the shortfall probabilities P [ X < e γ ] = P[à ] cannot decay at a faster rate than described by ii). Remark 4.3 Note that the constants α defined in part iii) of heorem 4. satisfy t log α = γ + log Q[Ac ] γ γ ), ) where we have used the crude estimate Q[A c ] e γ. Refined large deviation estimates for the probabilities Q[A c ] yield better rates for the convergence of α to. Indeed, Q[A c ]=Q[Z < e γ ] e ηγ E Q [Z η ] = e ηγ E P [Z η ] for any η>. he term E P [Z η ] can be computed explicitly; cf. Proposition 5.6. his yields with log Q[Ac ] γ + f η)) f η) := ρ σ ) η η) + η) 8 γ. 3

18 3 H. Föllmer, W. Schachermayer Sincewehaveassumedγ< σ 8 + ρ 4, the function f attains its maximum value σ f ηγ )) = 8 + ρ ) σ 4 γ 8 + ρ ) γ > ) in ηγ ) = ρ σ, ρ γ and so we can replace the rate γ γ in ) by the better rate γ γ + f ηγ )). Remark 4.4 he preceding discussion also applies to more general versions of the geometric Ornstein Uhlenbeck process. Suppose, for example, that the discounted price process is of the form S t = e Y t +µt with some constant drift parameter µ. In this case, the density Z of the unique equivalent martingale measure is again of the form 5), but with modified parameters σ = σ + µ σ, ρ = ρ + µ ) σ. As a result, Eq. 8) in Proposition 4. is still valid, but with σ instead of σ. Indeed, note that log Z takes the form 9) with ξ = η + ζ,where ζ = σ σ ρ Y Y ). he law of large numbers ) remains valid for ξ, and the large deviations principle for η can be transferred also to ξ. hus [ ] log P Z e γ = σ = I ρ log P [ η σ ρ )) 8 σ γ )] 8 σ γ for γ ], σ 8 + ρ [ 4, and this amounts to Eq. 8) with σ instead of σ. Finally we want to compare the arbitrage estimates for the geometric Ornstein Uhlenbeck case as described in heorem.5 and in Remark 4.4 with the much simpler case of the Black Scholes model. heorem 4.5 Let µ =, σ>and define the Black Scholes model by [ S t = S exp σ W t + µ σ ) ] t. ake γ ], ϕ [,whereϕ t ϕ µ σ. For any γ <γ, there exists X K such that, for any γ <γ γ, i) X e γ for large, ii) log P [ X < e γ ] ) = ϕ. γ ϕ More precisely, we can achieve the tighter bounds iii) X α := e γ Q[Ac ] Q[A ], 3

19 Asymptotic arbitrage and large deviations 3 where A ={Z e γ }.hen log α = γ γ ϕ + ϕ ), and the decay rate for the shortfall probabilities in ii) is optimal under the constraint iii). Proof For > the density Z of the unique equivalent martingale measure is given by ] Z = exp [ ϕw ϕ. We have P [ ] A c ] = P [ ϕw ϕ < γ ) ϕ = P W γ < ϕ ϕ = γ ) ) ϕ for ϕ>, and similarly for ϕ<, where denotes the distribution function of a standard normal variable. On the other hand, since Wt Q := W t + ϕt defines a Wiener process under Q, Q [ A c ] = Q [ ϕw Q < γ ϕ ] γ = ϕ + ϕ ) ). Now consider the contingent claim X := e γ A c α A. Clearly, X α and E Q [X ]=hence X K. Finally we have [ ] P X < e γ ϕ = P [A ] = γ ) ). ϕ Since ) log η = η for any η R, we easily obtain [ ] log P X < e γ ϕ = γ ) /. ϕ Inthesamewayweseethat log α = γ + log Q[Ac ] γ = γ ϕ + ϕ ). 3

20 3 H. Föllmer, W. Schachermayer As in the proof of heorem 4., optimality of the decay rate in ii) under the constraints iii) follows by applying the Neyman Pearson lemma. 5 Optimal strategies for the geometric Ornstein Uhlenbeck process for HARA utilities Let us return to the geometric Ornstein Uhlenbeck model introduced in Sect. 4. For three standard choices of a utility function U, we are going to compute the maximal expected utility u x) := E P [UX )] attainable at time > using some self-financing trading strategy and the initial capital x >. Recall see, e.g, [,]) that the optimal contingent claim X is of the form where the Lagrange multiplier y > isgivenby X = U ) yz ) ) E Q [ U ) yz ) ] = x. 3) Hence X x K. A little warning on the notation seems appropriate: While in the previous sections it was natural to have X K, we now follow the usual notation in utility maximization, where we fix an initial endowment x R and let X denote a random variable such that X x K. We are also going to identify the optimal trading strategy, i.e., the predictable process ξ ) t ) t such that X t ) := E Q [X F t ] t = x + ξ ) s ds s 4) for any t [, ]. It is clearly equivalent to compute the optimal proportion t := ξ t ) S t π ) of the capital X t ) generated up to time t which should be invested in the financial asset. Moreover, we are going to describe the growth of u x) and the iting form of the optimal strategy as increases to, and to give some financial interpretation in terms of certainty equivalents. 5. Logarithmic utility For the logarithmic utility function Ux) = log x it follows from ) and3) that the optimal contingent claim at time is given by X = xz. 6) hus the maximal expected utility takes the form 3 X ) t 5) u x) = log x + H P Q) 7)

21 Asymptotic arbitrage and large deviations 33 where [ ] [ ] H P Q) := E P log dp = E P log Z F dq denotes the relative entropy of P with respect to Q on F. Proposition 5. he maximal expected utility at time is given by u x) = log x + 4 ρ ) e ρ + ρ 8 4 σ y ) e ρ ) y e ρ + 8 σ. 8) In particular u x) grows linearly at the rate u x) = 4 ρ + 8 σ. 9) Proof In view of 7) it is enough to compute the relative entropy H P Q) = E P ρy t ) σ σ dw t + E P σ ρy t ) σ dt. Note that the first term vanishes, and recall that the first two moments of the Ornstein Uhlenbeck process Y t ) t are given by and hus E P [Y t ]=y e ρt 3) E P [Y t ]= σ ρ e ρt ) + e ρt y. H P Q) = ρ σ E P [Yt ]dt ρ E P [Y t ]dt + 8 σ = ρ ) 4 σ y e ρ ) y e ρ ) e ρ ρ + 8 σ. In the logarithmic case the optimal strategy does not depend on the horizon : Proposition 5. he optimal proportion defined by 5) is given by π t ) = σ ρy t. 3) Proof In view of 5), 6)and6) the optimal contingent claim is given by X = x exp ρy t ) σ σ dw t + σ ρy t ) σ dt = x exp ρy t ) σ σ dwt Q σ ρy t ) σ dt. 3

22 34 H. Föllmer, W. Schachermayer hus the Q-martingale satisfies X t ) := E Q [X F t ] t = x exp σ ρy s ) σ dws Q t σ dx t ) = X t ) ρy t )) σ σ dwt Q = X ) t σ ρy t + ) St ds t, ρy s ) σ ds and so the proportion π t ) defined by 5) is indeed given by 3). Remark 5.3 It is well-known that for logarithmic utility the optimal contingent claim is given by the numéraire portfolio X = x Z min ) = x + X t ϕ σ ) t ds t, 3) i.e., the optimal strategy is simply proportional to the current wealth X t as well as to the quotient ϕ σ ; see, e.g., [] and[4]. Note that these results also hold true in the general incomplete case. In our special case of the geometric Ornstein Uhlenbeck process, this is of course consistent with the explicit formula 3) for the optimal proportion. Formula 3) implies in particular that any estimate from below) on E [ log Z min)] yields an estimate from below) on E[logX )]. 5. Exponential utility For the exponential utility function Ux) = exp λx) with parameter λ>wehave U ) y) = λ log λ y ), so that the optimal contingent claim in ) takes the form X = x + λ H Q P) log Z ), where H Q P) := E Q [log Z ] denotes the relative entropy of Q with respect to P on F. Proposition 5.4 he maximal expected utility is given by u x) = exp λx H Q P)) 33) where ρ H Q P)= σ y ρ y + ) 8 σ + 4 ρ y )+ ) 8 ρσ + 4 ρ σ 3. 34) In particular, u x) grows to its upper bound at the rate 3 3 log u x)) = 4 ρ σ.

23 Asymptotic arbitrage and large deviations 35 Proof Since by ) [ u x) = E P exp λx ) ] = exp λx H Q P)), it remains to compute the relative entropy in our special setting. In view of 5)and6)we have H Q P) = E Q ρy t ) σ σ dw t σ ρy t ) σ dt = E Q ρy t ) σ σ dwt Q + σ ρy t ) σ ds = ρ σ E Q [Yt ]dt ρ E Q [Y t ]dt + 8 σ. Since Y t = σ W Q t and + y σ t by 6), the first two moments are given by E Q [Y t ]=y σ t. E Q [Yt ]=σ t + y y σ t + 4 σ 4 t, and this yields Eq. 34). Let us now identify the optimal strategy for a fixed horizon >. Proposition 5.5 he optimal quantity ξ t ) defined by 4) is given by ξ t ) = [ ρ + ρ λs t σ t) ) Y t ρ t)) + ]. 35) 4 Proof Consider the Q-martingale X t ) := E Q [X F t ] = x + λ H Q P) λσ t ρy s ) σ dws Q t λ λ E Q σ ρy s ) σ ds F t. t he last term can be computed explicitly using the conditional moments and E Q [Y s F t ]=σ W Q t E Q [Y s F t]=σ s t) + σ W Q t + y σ s ) + y σ s σ ρy s ) σ ds ) + σ Wt Q y ) σ s. 3

24 36 H. Föllmer, W. Schachermayer Since the components of bounded variation in our equation for the Q-martingale X t ) must sum up to the constant x, we finally obtain X ) t = x + λ = x + λ t t [ ) σ σ ρy s ρ σ s)y σ s 4 + σ ρ + ρ 4ρ s) ) ] dws Q S s [ ) ρσ ρ s) Y σ s ρ s)) + ] ds s. 4 his shows that the integrand ξ ) t ) defined via 4) is indeed given by 35). 5.3 Power utility Consider the power utility function Ux) = α xα with parameter α ], [\{}. Since U ) y) = y γ with γ := α ], [, the optimal contingent claim for > is given by X = xz γ E [ γ ] [ ] γ Q Z = xz E P Z β, 36) and the maximal expected utility takes the form u x) = E P [UX )] 37) u x) = xα [ ] α E P Z β α 38) where β := α ], [\{}. α he following proposition provides an explicit formula for u x). In particular it allows us to compute its rate of growth as. Note that for α>eq.4) describes the exponential growth of u x) to infinity, while for α<itspecifies the exponential decay of the distance between u x) and its maximal value. Proposition 5.6 We have E P [Z β ]=A ) exp B + A ) ) C, 39) where A ± := ) β ± exp ρ )) β, B := βy ρ ) [ β β) σ y 8 βσ + ρ ) ] β β), and C := y β exp ρ ) β + ρ ) y β β) σ exp ρ ) β + β σ 6 ρ β) exp ρ )) β. 3

25 Asymptotic arbitrage and large deviations 37 In particular, the maximal expected utility grows at the rate log u x) ) = 8 ασ + ρ α). 4) Proof In order to compute the expectation of Z β = exp ρβ σ Y s dy s + βρ σ Ys ds βy y ) 8 βσ, we first einate the energy term Y s ds appearing in the exponent by means of a suitable Girsanov transformation. For δ>wedenote by P δ the probability measure on F with density ϕ δ := exp ρ δ Y s dw s ) ρ δ Y s ds σ σ = exp ρ δ σ Y s dy s + ρ δ σ Ys ds 4) with respect to P.Since W δ t t := W t ρ δ Y s ds σ defines a Wiener process under P δ, Y t ) t becomes an Ornstein Uhlenbeck process under P δ with parameter δ, i.e., dy t = δy t dt + σ dwt δ. 4) Setting δ := ρ β and using Itô s formula Y s dy s = Y y ) σ, we obtain [ E P [Z β ]=Eδ Z β ) ] ϕ δ = E δ exp ρ ) β β) σ Y s dy s βy y ) 8 βσ ρ ) = expb ) E [exp δ β β) σ Y )] βy. In view of 4), Y is Gaussian with mean m := e δ y and variance ν := σ δ e δ ). Using the fact that E[expλY +ηy )]= λν ) exp λν ) λm +ηm+ )) η ν 43) 3

26 38 H. Föllmer, W. Schachermayer for any random variable Y with normal distribution Nm,ν ) and for λν <,wefinally obtain Eq. 39). Combining 38) with 39), we see that log u x) =α log x log α + α) B + A ) ) C α) log A. Since A and C converge to a finite it as, log u x) = α) B [ = α ) 8 βσ + ρ ) ] β β) = 8 ασ + ρ ) α. Our next goal is to identify the optimal strategy ξ t ) ) t defined by 4). It will be described in terms of the optimal proportion π t ) of the capital X t ) = E Q [X F t ] which should be invested in the financial asset at time t for any t [, ], see5). Proposition 5.7 he optimal proportion π t ) is an affine function of the logarithmic stock price given by π t ) = a t)y t + b t), 44) where a t) := ρ β A + σ t A t ) and b t) := [ A t ) β exp ρ )] β t). In particular, the asymptotic form of the optimal strategy for is given by π t := π ) ρ t = σ α Y t +. 45) Proof Consider the Q-martingale M t ) t defined by M t := E Q [Z γ F t], and recall the measure P δ introduced in the proof of Proposition 5.6 for δ := ρ β. In terms of its densities ϕt δ := E P [ϕ δ F t] with respect to P where ϕ δ is given by 4), M t takes the form ] M t = Zt E P [Z β F [ t ] = Zt ϕt δ Eδ Z β ϕδ ) F t [ = L t E δ ρ ) exp β β) σ Y ) ] βy F t 3

27 Asymptotic arbitrage and large deviations 39 with log L t := δ t σ Y s dy s δ t σ Y s ds + Y t y ) + 8 σ t + B where B was defined in Proposition 5.6. In view of 4) the random variable Y is Gaussian under the conditional distribution P δ [. F t ], with conditional mean m := Y t exp δ t)) and conditional variance ρ := σ δ exp δ t))). Using again formula 43) we finally obtain an expression of the form where M t = expn t )D t, N t := t π t ) σ dws Q is a Q-martingale, π t ) is given by 44)andD t ) t is some adapted process with continuous paths of bounded variation. But M t ) t is a Q-martingale, and this implies M t = M exp N t ) N t, hence and dx ) dm t = M t dn t t = X t ) dn t = X ) = X ) t π t ) t π t ) σ dw Q t S t ds t. hus we have shown that the trading strategy ξ t ) ) in 4) isgivenby ξ ) t = X ) t π t ) S t, and so the quantity π t ) defined by 44) is indeed the optimal proportion of the available capital X t ) which should be invested in the financial asset at time t. Since we obtain and A± = ) β, a t) = ρ ρ β = σ σ α b t) =, and so the asymptotic form of the strategy for is given by 45). 3

28 4 H. Föllmer, W. Schachermayer Remark 5.8 In Fleming and Sheu [3] dynamic programming methods are used in order to compute directly the optimal growth rate = sup sup log E P[UX π )] for power utilities U, where the supremum is taken over all admissible trading strategies; see also Pham [8]. Our Propositions 5.6 and 5.7 provide, in addition, explicit results for any finite horizon and may be viewed as a probabilistic complement to the analytical method in [3]. Remark 5.9 We have considered prices and contingent claims in discounted form since the bond was assumed to be normalized to B t. For a constant risk-free interest rate r >, the undiscounted contingent claim generated by a self-financing trading strategy is of the form X = X e r, and it seems natural to apply a given utility function U to X rather than to X. Moreover one may want to introduce a subjective rate of discounting δ>. Let us denoty by ũ x) the optimal value obtained by maximizing E [ U X )] e δ. For a power utility Ux) = α xα the optimal contingent claim is clearly the same as before, i.e., X = X e r where X is given by 36). Note, however, that Z now depends on r. he optimal value takes the form ũ x) = u x)e αr δ) where u x) is given by 37), and in analogy to Proposition 5.6 we obtain exponential growth at the rate log ũ x) ) = α r + ) ) σ σ r + ρ α ) δ; see [3] for detailed computations and for extensions to more general Ornstein Uhlenbeck processes in a robust setting. 5.4 Certainty equivalents and their growth rates We now shall interpret in financial terms the previous results on optimal investment with respect to the geometric Ornstein Uhlenbeck process for the utility functions considered above. We shall also analyze to which extent the logarithmic utility Ux) = logx) and the exponential utility Ux) = exp x) correspond to the iting cases α and α for power utilities U α) x) = xα α. o make the above results comparable we transform them from the utility scale to the money scale, using the concept of certainty equivalent, due to De Finetti compare, e.g., [7, Example 3.3.5]). We fix an initial endowment x of an economic agent, which we shall eventually normalize by x =, as well as the value function u x). he certainty equivalent CE x) then is the solution to Ux + CE x)) = u x). 46) he interpretation of this formula is that an agent whose preferences are modeled by expected utility U at time is indifferent between having an initial endowment x + CE x) 3

29 Asymptotic arbitrage and large deviations 4 without the possibility of investing in the financial market S so that her wealth remains constant during [, ]), as compared to having an initial endowment x as well as the possibility of investing optimally) in the market S during [, ]. By scaling we have CE x) = xce ) in the case of logarithmic and power utility while CE x) is independent of x in the case of the exponential utility. We therefore shall simply write CE for CE ); if we want to emphasize the role of the utility function U, weshall also denote this quantity by CE U. Propositions 5., 5.4 and 5.6 yield an explicit description of the growth of the certainty equivalents as Logarithmic utility We have CE log) x) = x exp H P Q) ), and formula 9) yields exponential growth at the rate log CElog) x) = ρ 4 + σ 8. 47) 5.4. Exponential utility We have CE exp) x) = λ H Q P), and formula 34) yields cubic growth at the rate Power utility 3 CEexp) x) = ρ σ λ 4. 48) For α ], [\{} and Ux) = xα α, the certainty equivalent is given by ] CE α) x) = x E P [Z β ) β, and formula 4) yields exponential growth at the rate log CEα) x) = σ 8 + ρ α). 49) α Remark 5. For the optimization problem formulated in Remark 5.9, the corresponding certainty equivalent CE x) is given by U x + CE x) ) = ũ x). For a power utility Ux) = α xα we obtain ) ) x + CE α) x) = x + CE α) x) r α e δ, 3

30 4 H. Föllmer, W. Schachermayer and 49) implies exponential growth at the rate t log CE α) x) = σ ) σ r + ρ α) + r δ α α. We now discuss the iting behavior α andα. We start with the first case which is easier. x Noting that α α α α = log x and α α = 4,wehavethat49) tends to 47)asα. In fact, a stronger result holds true: one may verify using 8), 38), 39)that, for fixed,wehave α CEα) = CE log). We thus see that α log CE α) ) = log CE log) ) = α log CE α) his formula indicates that the case of logarithmic utility indeed corresponds to the it α for power utility. We observe that the growth rate log CE α) ) of the certainty equivalent is a monotone function in α ], [ ranging from ) α log CE α) = σ 8 via ) α log CE α) ) = log CE log) = σ 8 + ρ 4 to ) α log CE α) = σ 8 + ρ. his reflects the financial intuition that an agent with smaller risk aversion can take better advantage measured in terms of certainty equivalents) of investment opportunities. he analysis of the behavior for α is more subtle than the case α. Recall that the exponential utility represents the it of U α) x) = xα α,asα, after proper affine normalisation: exp x) = x ) α x α) α = α α+. α α α α Hence, up to the multiplicative factor α α+, which is irrelevant for the certainty equivalent, the exponential utility exp x) is close to the shifted power utility U α) x α), as α. Using 34), 38), 39) one may verify that for fixed horizon,weagainhave α CEα) α ) = CEexp). 5) However, the above relation does not carry over to the iting expressions for by interchanging the iting procedures for α and as in the logarithmic case above: the terms in 48) and49) are of a completely different qualitative structure. In fact, the term 3 in 48) looks at first sight rather puzzling. 3 )

31 Asymptotic arbitrage and large deviations 43 In order to develop an understanding of the situation it is instructive to have a closer look at the optimal trading strategies. Let us again start with the logarithmic case which is the easiest, as π log, ) t = ρy σ t,asdefinedin3), does not depend on,sothatwemay define π log) t = σ ρy t independently of the horizon. As regards the optimal trading strategy π t α, ) the horizon ; nevertheless the it π α) and one readily verifies that π log) t = ρ σ Y t + = for power utility 44), it does depend on t for exists for each α ], [\{} 45) α ρ σ α Y t + = π t α), α in order words that the logarithmically optimal investment strategy again is the it of the iting α-optimal strategy π α),asα. Passing to the exponential utility, the picture changes drastically: the optimal quantity ξ exp, ) t,givenby35), depends on the horizon in such a way that it is not possible to pass to the it as the leading term is ρ 4λS t t). In order to find the reason why the exponential utility maximizer shows this rather strange behavior let us rewrite the value functions 7), 33)and38) in a form appropriate to argue with the Hamilton Jacobi-Bellman equation. u log) x, y, t) = log x + 4 ρ t) 8 u exp) x, y, t) = exp u α) xα x, y, t) = α where e ρ t)) + ρ 4 σ y e ρ t)) y e ρ t)) + 8 σ t), 5) [ ρ λx σ y ρ y + ) 8 σ t) 4 ρ y) + ) 8 ρσ t) ] 4 ρ σ t) 3, 5) [ A t) exp B t + A ) ) ] α t C t, 53) ) β ± exp ρ )) β t), ρ ) β β) σ y A ± t := B t := βy [ 8 βσ + ρ ) ] β β) t), and C t := ρ yβ exp ) β t) + ρ ) y β β) σ exp ρ ) β t) + β σ 6 ρ β) exp ρ )) β t). A basic feature of dynamic programming is that, fixing and plugging the optimizer X t ) ) t as well as the process Y t) t into the value function above, one obtains a local) martingale u X t ), Y t, t )). We do not try to prove this rigorously which t 3

32 44 H. Föllmer, W. Schachermayer is possible) as we only want to argue formally to develop an intuition for the phenomena encountered above. We shall also use the formal identities E[dW t ]=anddw t ) = dt. We obtain from the martingale property the equation E [ d u X ) t where Itô s formula allows us to write [ ))] E d u X t ), Y t, t = u dt + u [ t x E d X t ) ))], Y t, t =, 54) ] + u y E [dy t] + u x d X t ) + u y dy t) + u x y d X ) t dy t 55) Analyzing the above equation, it turns out that the mixed derivative u x y x, y, t) plays a crucial role: while this term vanishes in the case of logarithmic utility 5), it cannot be neglected in the exponential 5) and the power case 53). Concentrating on the exponential case we deduce from 5) that the leading term w.r. to t)of u exp) x y is given by ) u exp) x y λρ 4 t) u exp). his term dominates for large t) the derivatives uexp) x and λ u exp) respectively. λu exp) as well as u exp), which equal x his gives us a clue to the understanding of the leading term of the optimal strategy ξ t ) ρ t). 56) 4λS t We may interpret the dynamic programming equation 54) in the following way: the exponential utility maximizer chooses at time t the investment ξ t ) in such a way that, when plugging d X t ) = ξ t ) ds t into 54) and55) theterm u t becomes minimal. Indeed, in this case the function u.,.,) becomes maximal for the given boundary condition u x, y, ) = exp λx) as the descent in the time coordinate is steepest. he choice of the control variable ξ t ) in d X t ) = ξ t ) ds t does not effect the behavior of E[dY t ] or dy t ) ; hence, we see from 55) that one has to solve the following maximization problem for the variable ξ R: and Using 3 u x ξe[ds t]+ u x ξ ds t ) + u x y ξds tdy t max! u x = λu, u x = λ u, u x y λρ 4 t) u σ E[dS t ]=e y ) ρy dt, ds t ) = e y σ dt, ds t dy t = σ e y dt

33 Asymptotic arbitrage and large deviations 45 we arrive, letting S t = e y and keeping only the leading terms in t),at λ u ξ e y σ dt λρ 4 t) u ξσ e y dt max! Differentiating with respect to ξ and equating to zero yields 56). Summing up, we see that in the exponential case the joint wobbling d X t ) mixed derivative u x y are the crucial terms which cause the optimal value ξ ) t S t = ρ 4λ dy t and the t) to be of the order t), independently of the current value of the process Y t which determines the market price of risk. Now we are in a position to understand why the double iting procedure α, cannot be interchanged: For fixed horizon we have that u α) x α) tends to u exp) x), asα for x R, and the same iting behavior holds true for the corresponding trading strategies. On the other hand the optimal exponential strategies ξ t ) 56) do not converge, as, while we have seen that, for fixed α [, [\{} the optimal power strategies π t α, ) do converge to π t α). Not only in the case of exponential utility we find remarkable differences to the Black Scholes situation where the optimal strategies are independent of the time horizon, but even in the case of logarithmic utility, which is the most regular one, we find unexpected phenomena. As regards the optimal investment of the logarithmic utility maximizer there are no surprises yet: the optimal investment proportion π log) t given in 3) is proportional to the market price of risk ρ σ Y t + σ of the process S see 4)). When Y t = σ ρ there are no profitable investment opportunities as in this case the market price of risk vanishes: the optimal strategy of the log-optimizer therefore is not to invest into the risky asset in this situation. Now let us look at the value function 5) for the logarithmic utility u log) x, y, t) = log x + 4 ρ t) ρ e t)) 8 + ρ 4 σ y e ρ t)) y e ρ t)) + 8 σ t). Fixing x and t we observe that the minimal value is attained at y min σ ρ for large. Hence the situation for the log-investor is worse if measured by expected utility at time ) if Y t = σ ρ than in the case Y t = σ ρ, when the market price of risk vanishes. o explain this at first glance counter-intuitive phenomenon we quote from Wilhelm ell: He has no choice but through this sunken way to come to Kussnacht. here is no other road. he drift of the Ornstein Uhlenbeck process Y drives the process back towards Y = ; hence a typical path of Y, for which at time t we have Y t = σ ρ, will tend towards and on this way it has to pass through the region around σ ρ where the log-investor will not find profitable investment opportunities. his situation is worse than starting at Y t = σ ρ, where one already may hope to drift out of the sunken way. Let us finally look at the pathwise growth of the capital X generated by the optimal strategy up to time. Note first that ρ log Z = 4 + σ ) P-a.s., 57) 8 due to 9)and). 3